Expressivity of Determinantal Ansatzes for Neural Network Wave Functions

Neural network wave functions have shown promise as a way to achieve high accuracy in solving the many-body quantum problem. These wave functions most commonly use a determinant or a sum of determinants to antisymmetrize many-body orbitals, which are described by a neural network. In many cases, the wave function is projected onto a fixed-spin state. Such a treatment is allowed for spin-independent operators; however, it cannot be applied to spin-dependent problems, such as Hamiltonians containing spin-orbit interactions. We show that for spin-independent Hamiltonians, a strict upper bound property is obeyed between a traditional Hartree-Fock-like determinant, full spinor wave function, the full determinant wave function, and a generalized spinor wave function. The relationship between a spinor wave function and the full determinant arises because the full determinant wave function is the spinor wave function projected onto a fixed-spin, after which antisymmetry is implicitly restored in the spin-independent case. For spin-dependent Hamiltonians, the full determinant wave function is not applicable, because it is not antisymmetric. Numerical experiments on the H₃ molecule and two-dimensional homogeneous electron gas confirm these bounds.